Introduction

Once one has obtained a solid understanding of the fundamentals of
Fourier series
analysis and the
General Derivation of the Fourier Coefficients , it is useful to have an understanding of the common signals used in Fourier Series Signal Approximation.

Deriving the coefficients

Consider a square wave f(x) of length 1. Over the range [0,1), this can be written as

x(t)=1t≤12;-1t>12.

Fourier series approximation of a square wave

Fourier series approximation to
sqt .
The number of terms in the Fourier sum is indicated in eachplot, and the square wave is shown as a dashed line over two
periods.

Real even signals

Given that the square wave is a real and even signal,

f(t)=f(-t) EVEN

f(t)=f *
(t) REAL

therefore,

cn=c-n EVEN

cn=cn * REAL

Deriving the coefficients for other signals

The Square wave is the standard example, but other important signals are also useful to analyze, and these are included here.

Constant waveform

This signal is relatively self-explanatory: the time-varying portion of the Fourier Coefficient is taken out, and we are left simply with a constant function over all time.

x(t)=1

Fourier series approximation of a constant wave

Sinusoid waveform

With this signal, only a specific frequency of time-varying Coefficient is chosen (given that the Fourier Series equation includes a sine wave, this is intuitive), and all others are filtered out, and this single time-varying coefficient will exactly match the desired signal.

x(t)=cos(2πt)

Fourier series approximation of a sinusoid wave

Triangle waveform

x(t)=tt≤1/21-tt>1/2

This is a more complex form of signal approximation to the square wave. Because of the
Symmetry Properties of the Fourier Series, the triangle wave is a real and odd signal, as opposed to the real and even square wave signal. This means that

f(t)=-f(-t) ODD

f(t)=f *
(t) REAL

therefore,

cn=-c-n

cn=-cn * IMAGINARY

Fourier series approximation of a triangle wave

Sawtooth waveform

x(t)=t/2

Because of the
Symmetry Properties of the Fourier Series, the sawtooth wave can be defined as a real and odd signal, as opposed to the real and even square wave signal. This has important implications for the Fourier Coefficients.

Fourier series approximation of a sawtooth wave

Dft signal approximation

Interact (when online) with a Mathematica CDF demonstrating the common Discrete Fourier Series. To download, right-click and save as .cdf.

Conclusion

To summarize, a great deal of variety exists among the common Fourier Transforms. A summary table is provided here with the essential information.

Common discrete fourier transforms

Description

Time Domain Signal for
n∈Z[0,N-1]

Frequency Domain Signal
k∈Z[0,N-1]

Constant Function

1

δ(k)

Unit Impulse

δ(n)

1N

Complex Exponential

ej2πmn/N

δ((k-m)N)

Sinusoid Waveform

cos(j2πmn/N)

12(δ((k-m)N)+δ((k+m)N))

Box Waveform
(M<N/2)

δ(n)+∑m=1Mδ((n-m)N)+δ((n+m)N)

sin((2M+1)kπ/N)Nsin(kπ/N)

Dsinc Waveform
(M<N/2)

sin((2M+1)nπ/N)sin(nπ/N)

δ(k)+∑m=1Mδ((k-m)N)+δ((k+m)N)

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?

fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.

Tarell

what is the actual application of fullerenes nowadays?

Damian

That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.

Tarell

Join the discussion...

what is the Synthesis, properties,and applications of carbon nano chemistry

Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.